reserve X for set;

theorem Th8:
  for X be non empty set holds for x,y be Element of InclPoset X st
  x \/ y in X holds x "\/" y = x \/ y
proof
  let X be non empty set;
  let x,y be Element of InclPoset X;
  assume x \/ y in X;
  then reconsider z = x \/ y as Element of InclPoset X;
  y c= z by XBOOLE_1:7;
  then
A1: y <= z by Th3;
A2: now
    let c be Element of InclPoset X;
    assume x <= c & y <= c;
    then x c= c & y c= c by Th3;
    then z c= c by XBOOLE_1:8;
    hence z <= c by Th3;
  end;
  x c= z by XBOOLE_1:7;
  then x <= z by Th3;
  hence thesis by A1,A2,LATTICE3:def 13;
end;
